UNIVERSITAT DE BARCELONA

Transcription

1 UNVERSTAT DE BARCELONA LOCAL COHOMOLOGY MODULES SUPPORTED ON MONOMAL DEALS by Josep Àlvarez Montaner Departament d Àlgebra i Geometria Facultat de Matemàtiques Març 2002

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3 UNVERSTAT DE BARCELONA LOCAL COHOMOLOGY MODULES SUPPORTED ON MONOMAL DEALS Memòria presentada per Josep Àlvarez Montaner per a aspirar al grau de Doctor en Matemàtiques Departament d Àlgebra i Geometria Facultat de Matemàtiques Març 2002

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5 Departament d Àlgebra i Geometria, Universitat de Barcelona Programa de Doctorat d Àlgebra i Geometria Doctorand: Josep Àlvarez Montaner Tutor: Santiago Zarzuela Armengou Director de Tesi: Santiago Zarzuela Armengou Santiago Zarzuela Armengou Professor Titular del Departament d Àlgebra i Geometria de la Universitat de Barcelona CERTFCA: Que la present memòria ha estat realitzada sota la seva direcció per Josep Àlvarez Montaner i que constitueix la tesi d aquest per a aspirar al grau de Doctor en Matemàtiques. Barcelona, Març de 2002 Signat: Santiago Zarzuela Armengou.

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7 Als meus pares i germans

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9 Contents ntroduction v 1 Preliminaries Local Cohomology Modules Squarefree monomial ideals Z n -grading Free resolutions njective resolutions Further considerations Hilbert series Stanley-Reisner ring Some particular posets Arrangement of linear subvarieties D-Modules Ring of differential operators nverse and direct image Further considerations on D-module theory Geometry of the characteristic cycle Some examples and computations i

10 ii Solutions of a D X -module Riemann-Hilbert correspondence Some Numerical nvariants of Local Rings Lyubeznik numbers Lyubeznik numbers and characteristic cycles Multiplicities of the Characteristic Cycle Multiplicities of CC(H n i (R)) Multiplicities of CC(H p p (H n i (R))) Characteristic cycle of local cohomology modules Cohen-Macaulay Case Main Result Mayer-Vietoris process Mayer-Vietoris process and characteristic cycle Optimization of the Mayer-Vietoris process Main result Characteristic cycle and restriction Consequences Annihilation and support of local cohomology modules Arithmetical properties Combinatorics of the Stanley-Reisner ring and multiplicities Betti numbers and multiplicities Bass numbers of local cohomology modules Cohen-Macaulay Case Main result

11 iii Double process Double process and characteristic cycle Optimization of the process Main result Consequences Bass numbers of local cohomology modules Bass numbers and restriction Some particular cases njective dimension of local cohomology modules Associated prime ideals of local cohomology modules Small support of local cohomology modules Local cohomology, arrangements of subspaces and monomial ideals Filtrations on local cohomology modules The Roos complex The Roos complex of an arrangement of linear varieties Characteristic cycle of local cohomology modules supported on arrangements of linear varieties Straight modules Extension problems Modules with variation zero The category of n-hypercubes The functor GGM Modules with variation zero Projective, njective and Simple modules An Equivalence of Categories

12 iv The graded structure of D X -modules with variation zero Local cohomology modules Topological interpretation Resum en català ntroducció Objectius Conclusions Bibliography 229

13 ntroduction The aim of this work is the study of local cohomology modules supported on monomial ideals. Let us first introduce the subject and main problems. After this we will expose some known related results and, finally, we will give a summary of the results obtained in this work. Local cohomology was introduced by A. Grothendieck as an algebraic cohomology theory analogous to the classical relative cohomology. This analogy comes from the fact that many results on projective varieties can be reformulated in terms of graded rings or complete local rings. The notes published by R. Hartshorne based on a course given by A. Grothendieck [38] will be the starting point for introducing this theory. Let X be a topological space. Given a locally closed subspace Z X we consider the functor of sections with support in Z, that will be denoted by Γ Z (X, F) where F is a sheaf of abelian groups over X. This is a left exact functor. Since the category of sheaves of abelian groups on X has enough injectives, one may consider the right derived functors of Γ Z (X, F) that will be called local cohomology groups of X with coefficients in F and support in Z, and will be denoted by HZ r (X, F) := Rr Γ Z (X, F). A first interpretation of these groups is given by the existence of the long exact sequence of cohomology groups 0 H 0 Z (X, F) H 0 (X, F) H 0 (U, F U ) H 1 Z (X, F) where U = X \ Z is the complement of Z in X. Then, HZ 1 (X, F) is nothing but the obstruction to the extension of sections of F over U to the whole space X. Even though the geometric point of view of the notes by A. Grothendieck, local cohomology quickly became an indispensable tool in the theory of Noethev

14 vi ntroduction rian commutative rings. n particular, R. Y. Sharp [83], described local cohomology in the framework of Commutative Algebra. Throughout this work, we will consider the following situation: Let R be a Noetherian commutative ring. Given an ideal R we consider the functor of -torsion, that is defined for any R-module M as Γ (M) := {x M n x = 0 for any n 1}. This is an additive, covariant and left exact functor. Since the category of R-modules has enough injectives, one may consider the right derived functors of Γ that will be called local cohomology modules of M with support the ideal and will be denoted by H r (M) := R r Γ (M). Whether R is Noetherian, the sheaf associated to an injective R-module is flasque. From this fact, one may check that the notion of local cohomology defined by using the theory of sheaves coincide with the local cohomology defined in algebraic terms. More precisely, let X = Spec R be an affine space, F = M the sheaf associated to a R-module M and Z the subvariety defined by an ideal R. Then, one has isomorphisms HZ r (X, M) = H r (M) for all r. Despite the effort of many authors in the study of these modules, their structure is still quite unknown. Following C. Huneke s criteria [45], the basic problems concerning local cohomology modules are: Annihilation of local cohomology modules. Finitely generation of local cohomology modules. Artinianity of local cohomology modules. Finiteness of the associated primes set of local cohomology modules. n general, one can not even say when they vanish. Moreover, when they do not vanish they are rarely finitely generated. However, in some situations these modules verify some finiteness properties that provide a better understanding of their structure. Next we will expose some results one may find in the literature related to these problems and are best related with the contents of this work. Moreover, we will comment some applications both in Algebraic Geometry and Commutative Algebra.

15 ntroduction vii Annihilation of local cohomology modules A. Grothendieck obtained the first basic results on annihilation by giving bounds for the possible integers r such that H r (M) 0 in terms of the dimension and the grade. Namely, let M be a finitely generated R-module, then one has grade (, M) r dimr for all r such that H r (M) 0. Whether (R, m) is a Noetherian local ring, this result may be completed in order to provide a cohomological characterization of the Krull dimension. Namely, H r m(m) vanishes for all r > dimm and it is different from zero for r = dimm. Even though the lower bound is sharp, it does not occur for the upper bound. n order to precise this bound, we introduce the integer cd (R, ) := max{r H r (M) 0 M}, that will be named the cohomological dimension of the ideal with respect to R. t is worthwhile to point out that one only has to study the case M = R since the cohomological dimension equals to the biggest integer such that H r (R) 0. Among the results that we may found on the annihilation of local cohomology modules we detach the Hartshorne-Lichtenbaum vanishing theorem [40]: Theorem Let (R, m) be a complete local domain of dimension d. R be an ideal, then cd (R, ) < d if and only if dimr/ > 0. The following theorem was proved by R. Hartshorne [40] in the geometric case, by A. Ogus [74] in characteristic zero and by C. Peskine-L. Szpiro [76] and R. Hartshorne-R. Speiser [43] in characteristic p > 0. A characteristic free proof was given by C. Huneke-G. Lyubeznik [47]. Theorem Let (R, m) be a complete local domain of dimension d with separately closed residue field. Let R be an ideal, then cd (R, ) < d 1 if and only if dimr/ > 1 and Spec (R/) \ {m} is connected. Unfortunately, there is not a simple extension of these result to lower cohomological dimensions. However, one may found bounds for the cohomological dimension in some special cases. From this point of view, we detach the work of G. Faltings [24] and Huneke-Lyubeznik [47]. To illustrate the utility of local cohomology modules we are going to announce some applications of the results described above. Let

16 viii ntroduction Let ara() be the minimum number of generators that are required in order to define the ideal but radical. Then, one has cd (R, ) ara(). We have to point out that this integer has a key role in the study of the connectivity of algebraic varieties (see [15]). n the case of subvarieties of projective spaces, the local cohomology annihilation theorems have topological applications (see [40], [74], [47]). For example, one may obtain the following generalization of the Lefschetz theorem: Let R = C[x 0,..., x n ] be the defining ideal of a closed subvariety Y P n C. Then, the morphism H r dr(p n C) H r dr(y ), where HdR r ( ) denote the de Rham cohomology groups, is an isomorphism for all r < n cd (R, ) and is a monomorphism for r = n cd (R, ) ([42, Theorem.7.1]). Finitely generation of local cohomology modules n general, the modules H r (M) are rarely finitely generated, even if the module M is. G. Faltings [23] gave a criteria to determine the finitely generation of local cohomology modules. This criteria depends on the numbers s(, M) := min{depth (M p ) + ht (( + p)/p) p, p Spec (R)}. Theorem Let R be a Noetherian ring, R an ideal and M a finitely generated R-module. Then, H r (M) is finitely generated for all r < s(, M) and is not for r = s(, M). A. Grothendieck conjectured that even though the local cohomology modules H r(r) are not finitely generated, the modules Hom R(R/, H r (R)) are. R. Hartshorne showed that this conjecture is false [41]. However, there has been a big effort in the study of the cofiniteness of local cohomology modules, where we say that a R-module M is -cofinite if Supp R (M) V () and Ext r R(R/, M) is finitely generated for all r 0. From this point of view, we detach the work of Huneke-Koh [46], D. Delfino [18] and Delfino-Marley [19]. Finiteness properties of local cohomology modules Although the local cohomology modules are in general not finitely generated, under certain conditions they satisfy some finiteness properties that

17 ntroduction ix provide a better understanding of their structure. remak the following result: Following this path, we Let R be a unramified regular local ring. Then, for any ideal R, any prime ideal p R and any r 0 one has: The set of associated primes of H r (R) is finite. The Bass numbers µ p (p, H r (R)) are finite. This result has been proved by Huneke-Sharp [48] in the positive characteristic case and by G. Lyubeznik in the zero characteristic [55] and mixed characteristic case [57]. We have to point out that G. Lyubeznik uses the algebraic theory of D- modules due to the fact that local cohomology modules are finitely generated as D-modules. R. Hartshorne [41] gave an example of a local cohomology module whose Bass numbers may be infinite if R is not regular. The finiteness of the associated primes set of H r (R) for any Noetherian ring R and any ideal was an open question until A. Singh [84] (non local case) and M. Katzman [51] (local case) have given examples of local cohomology modules having infinite associated primes. By using the finiteness of Bass numbers, G. Lyubeznik defined a new set of numerical invariants for local rings A containing a field, that are denoted as λ p,i (A). More precisely: Let (R, m, k) be a regular local ring of dimension n containing a field k, and let A be a local ring that admits an epimorphism of rings π : R A. Let = Ker π. Then one defines λ p,i (A) as the Bass number µ p (m, H n i (R)) = dim k Ext p R R nor on π. (k, Hn i (R)). These numbers depend on A, i and p, but neither on They have an interesting topological interpretation, as it was pointed out by G. Lyubeznik. Let V be a scheme of finite type over C of dimension d and let A be the local ring of V at a closed point q V. f q is an isolated singular point of V then, from a theorem of A. Ogus [74] relating local cohomology and algebraic de Rham cohomology, and the comparison theorem between algebraic de Rham

18 x ntroduction cohomology and singular cohomology proved by R. Hartshorne in [42], one gets λ 0,i (A) = dimc H i q(v, C) for 1 i d 1, where H i q(v, C) is the i-th singular local cohomology group of V with support in q and with coefficients in C. R. Garcia and C. Sabbah [29] generalize this result for the pure dimensional case by using the theory of D-modules. n particular, they express these Lyubeznik numbers in terms of Betti numbers of the associated real link. Graded structure of local cohomology modules Whether the ring R and the ideal are graded, the local cohomology modules H r (M) have a graded structure too for any graded R-module M. We have to point out that the graded version of the main results on local cohomology modules remain true (see [15]). Graded local cohomology has interesting applications in projective Algebraic Geometry. n particular, the Castelnuovo-Mumford regularity reg (M), is an invariant of the R-module M, determined by the graded local cohomology. Whether M is a finitely generated module, this invariant provides some information on the resolution of the module. For example, let M = P n (V ) be k the defining ideal of a projective variety V P n k, where k is an algebraically closed field. Then, the grades of the homogeneous polynomials that define (V )). P n k (V ) can not be larger than reg ( P n k Some different applications of graded local cohomology may be found in the study of graded rings associated to filtrations of a commutative ring R, mainly the Rees algebra and the associated graded ring of an ideal R. Recall that these rings play an essential role in the study of singularities, since they become the algebraic realization of the classical notion of blowing up along a subvariety. Local cohomology modules supported on monomial ideals Let R = k[x 1,..., x n ], where k is a field, be the polynomial ring on the variables x 1,..., x n. Let m := (x 1,..., x n ) R be the homogeneous maximal ideal and R a squarefree monomial ideal. Then, the local cohomology modules H r m(r/) and H r (R) have a natural Zn -structure, where the corresponding graded pieces are k-vector spaces of finite dimension.

19 ntroduction xi From the Taylor resolution of the quotient ring R/, G. Lyubeznik [54] gave a description of the local cohomology modules H r (R), determined their annihilation and described the cohomological dimension of with respect to R. By the Stanley-Reisner correspondence, one may associate to any squarefree monomial ideal R a simplicial complex defined on the set of vertices {x 1,..., x n }. n [86] one may found a result of M. Hochster where a description of the graded Hilbert series of the local cohomology modules Hm(R/) r is given in terms of the reduced simplicial cohomology of certain subcomplexes of. More precisely, given a face σ α := {x i α i = 1}, we define: Link of σ α in : link α := {τ σ α τ =, σ α τ }. Restriction to σ α : α := {τ τ σ α }. Let be the Alexander dual simplicial complex of. Then, from the equality of complexes 1 α = (link α ) and Alexander duality we get the isomorphisms H n r α 1 (link α ; k) = H r 2 ( 1 α; k). Moreover, we have to point out that the inclusion 1 α ε i the morphisms: 1 α induces H n r α εi 1(link α εi ; k) H n r α 1 (link α ; k), M. Hochster s result is then as follows: Theorem The graded Hilbert series of H r m(r/) is: H(H r m(r/); x) = σ α dim k Hr α 1 (link α ; k) α i =1 x 1 i 1 x 1 i. From M. Hochster s formula one may deduce that the multiplication by the variable x i establishes an isomorphism between the pieces Hm(R/) r β and Hm(R/) r β+εi for all β Z n such that β i 1, where ε 1,..., ε n is the natural basis of Z n. Notice then that, in order to determine the graded structure of this module, we only have to determine the multiplication by x i on the pieces Hm(R/) r α, α {0, 1} n.

20 xii ntroduction H. G. Gräbe [35], gave a topological interpretation of these multiplications by using the isomorphisms: H r m(r/) β = Hr α 1 (link α ; k), β Z n where sup (β) := {x i β i < 0}. such that σ α = sup (β), Theorem For all α {0, 1} n such that σ α, the morphism of multiplication by the variable x i : corresponds to the morphism x i : H r m(r/) α H r m(r/) (α εi ) H r α 1 (link α ; k) H r α ε i 1 (link α εi ; k), induced by the inclusion 1 α ε i 1 α. nspired by M. Hochster s formula, N. Terai [92] gave a description of the graded Hilbert series of the local cohomology modules H r (R), expressed in terms of the reduced simplicial homology of the links link α such that σ α, α {0, 1} n. Theorem The graded Hilbert series of H r (R) is: H(H r (R); x) = dim k Hn r α 1 (link α ; k) α {0,1} n α i =0 x 1 i 1 x 1 i α j =1 1 1 x j. From N. Terai s formula one also may deduce that the multiplication by the variable x i establishes an isomorphism between the pieces H r(r) β and H r(r) β+ε i for all β Z n such that β i 1. ndependently, M. Mustaţă [72] has also described the pieces of the local cohomology modules H r (R) and, moreover, he has given a topological interpretation of the morphism of multiplication by x i on the pieces H r(r) α, α {0, 1} n. We have to point out that these results have been used for the computation of cohomology of coherent sheaves on toric varieties (see [22]). Theorem For all α {0, 1} n such that σ α, the morphism of multiplication by the variable x i : x i : H r (R) α H r (R) (α εi )

21 ntroduction xiii corresponds to the morphism H n r α 1 (link α ; k) H n r α εi 1(link α εi ; k), induced by the inclusion 1 α ε i 1 α. We remark that the formulas of M. Hochster and N. Terai are equivalent by using the Čech hull and Alexander duality (see [70]). The same happens with the formulas of H. G. Gräbe and M. Mustaţă. Finally, K. Yanagawa [97] has introduced the category of straight modules, that are Z n -graded modules such that the multiplication by the variables x i between their pieces satisfy certain conditions. n this framework, K. Yanagawa can study the local cohomology modules since, from the results of N. Terai and M. Mustaţă one may check that the shifted modules H r (R)( 1,..., 1) are straight. Algorithmic computation of local cohomology modules Let k be a field of characteristic zero, R = k[x 1,..., x n ] the ring of polynomials over k in n variables and D be the corresponding ring of differential operators. Recently, it has been a great effort to provide an effective computation of local cohomology modules. F. Barkats [3] gave an algorithm to compute a presentation of local cohomology modules H r (R) supported on monomial ideals R by using the Taylor resolution of the quotient ring R/. This algorithm was implemented for ideals contained in R = k[x 1,..., x 6 ]. By using the theory of Gröbner bases over the ring D we can find two different methods for computing local cohomology modules. The first one is due to U. Walther [93] and is based on the construction of the Čech complex of holonomic D-modules. n particular, let m be the homogeneous maximal ideal and R be any ideal. Then, U. Walther determines the structure of the modules H r(r), Hp m(h r(r)) and compute the Lyubeznik numbers λ p,i(r/). The second method is due to T. Oaku and N. Takayama [73]. t relies in their algorithm for computing the derived restriction modules of holonomic D-modules. Computations can be done in the computer algebra system Macaulay 2 [37] by using the package D-modules for Macaulay 2 [53].

22 xiv ntroduction Objectives n the sequel, we are going to introduce the problems considered in this work. Let R be a regular ring containing a field of characteristic zero and R an ideal. Our intend is, following the path opened by G. Lyubeznik in [55], to make a deeper use of the theory of D-modules in order to study the local cohomology modules H r (R). We are mainly interested on an effective description of the annihilation and finiteness properties of these modules. The main tool we will use is the characteristic cycle. This is an invariant that one may associate to any holonomic D-module M (e.g. local cohomology modules) which, in the cases we will consider in this work, is described as a sum CC(M) = m i T X i X, where m i Z, X = Spec (R) and T X i X is the conormal bundle relative to the subvariety X i X. Notice that the information given by this invariant is a set of subvarieties X i X and a set of multiplicities m i. The usefulness of the characteristic cycle in our study is reflected in the following examples: The support as R-module of a D-module M is described by the subvarieties that appear in the characteristic cycle. The Lyubeznik numbers λ p,i (R/) may be computed as the multiplicities of the characteristic cycle of the module H p m(h r (R)). More precisely, let R = k[[x 1,..., x n ]] be the formal power series ring with coefficients in a field k of characteristic zero. Let m = (x 1,..., x n ) R be the maximal ideal and R be any ideal. Then, from the results of [55], one may see that H p m(h r (R)) = H n m(r) λ p,i(r/), where λ p,i (R/) are the Lyubeznik numbers. By computing the characteristic cycle, one may check that these invariants of the quotient ring R/ are nothing but the corresponding multiplicities. Observe then that the characteristic cycles of the local cohomology modules H r(r) and Hp m(h r (R)) provide information for both the module and the quotient ring R/ as well.

23 ntroduction xv Throughout this work, we will be interested in the interpretation of the information given by the characteristic cycle and make explicit computations. The main problems we want to treat are: Problem 1. Let be an ideal contained in a regular local ring R. Then, we want to study the invariance of the multiplicities of the characteristic cycle of the local cohomology modules H r (R) with respect to the quotient ring R/. Let R be the polynomial ring or the formal power series ring with coefficients in a field of characteristic zero and R be a squarefree monomial ideal. Recall that these ideals can be interpreted in the following ways: Stanley-Reisner ideal of a simplicial complex. Defining ideal of an arrangement of linear varieties. Then, the problems we will consider in this case are: Problem 2. Explicit computation of the characteristic cycle of local cohomology modules. From this computation we will turn our interest in the study of: Study the support of local cohomology modules. n par- Problem 2.1 ticular: Annihilation of local cohomology modules. Cohomological dimension. Description of the support of local cohomology modules. Krull dimension of local cohomology modules. Artinianity of local cohomology modules. Problem 2.2 Arithmetical properties of the quotient rings R/. Namely, we want to determine the following properties: Cohen-Macaulay property. Buchsbaum property. Gorenstein property. The type of Cohen-Macaulay rings.

24 xvi ntroduction Problem 2.3 nterpretation of the multiplicities of the characteristic cycle of the local cohomology modules H r (R). n particular: Study the topological and algebraic invariants of the Stanley-Reisner simplicial complexes associated to the rings R/ and R/, where is the Alexander dual ideal of. Study the topological invariants of the complement of the arrangement of linear varieties defined by the ideal. Problem 2.4 Explicit computation of the Bass numbers of the local cohomology modules. From these computations we will consider: Annihilation of Bass numbers. njective dimension of local cohomology modules. Associated primes of local cohomology modules. Small support of local cohomology modules. Even though it provides many information, the characteristic cycle does not describe completely the structure of the local cohomology modules. This fact is reflected in the work of A. Galligo, M. Granger and Ph. Maisonobe [27], where a description of the category of regular holonomic D-modules with support a normal crossing (e.g. local cohomology modules supported on squarefree monomial ideals), is given by using the Riemann-Hilbert correspondence. Then, we will also consider the following question: Problem 3. Study the structure of local cohomology modules by using the following points of view: Study the Z n -graded structure of local cohomology modules. Study the regular holonomic D-modules with support a normal crossing and Riemann-Hilbert correspondence.

25 ntroduction xvii Conclusions The contents of this thesis are the following: n Chapter 1 we introduce the definitions and notation we will use throughout this work. n a first section we give the definition of local cohomology module and we state some of the basic properties we will use. Finally, we introduce some of the tools that will allow us to compute these modules: The Čech complex, the long exact sequence of local cohomology, the Mayer-Vietoris sequence, the Brodmann s sequence and the Grothendieck s spectral sequence. n a second section we are focused in the study of squarefree monomial ideals. A natural framework for these ideals is the category of Z n -graded modules. s for this reason that we recall the notions of free and injective resolutions in this category. Recently, it has been a big effort in the study of these resolutions in order to make them treatable, so have found convenient to introduce the notions of cel.lular matrices, cel.lular resolutions and Čech hull. To any squarefree monomial ideal one can associate a simplicial complex by means of the Stanley-Reisner correspondence. For this reason we review some topological notions, with an especial attention to Alexander duality. On the other side we can refer to any squarefree monomial ideal as the defining ideal of an arrangement of linear subvarieties. From this point of view, we recall Goresky-MacPherson s formula for the computation of the cohomology of the complement of these type of arrangements. We also review the theory of D-modules. We start giving the basic definitions of regular holonomic D-modules since local cohomology modules are of this type. To these modules one may associate an invariant, the characteristic cycle, that allows us to compute the support of these modules. After describing some geometric properties of these characteristic cycles, we develop some examples and computations. Finally, we introduce the notion of solution of a D-module. We recall that the solutions of a regular holonomic D-module are Nilsson class functions and that the solutions functor restricted to the category of regular holonomic D- modules establishes a categorical equivalence with the category of perverse sheaves that is named the Riemann-Hilbert correspondence.

26 xviii ntroduction n Chapter 2 we prove that the multiplicities of the characteristic cycle of local cohomology modules are invariants of the corresponding quotient ring. n particular, these invariants generalize Lyubeznik numbers. Given a field k of characteristic zero we consider the ring of formal power series R = k[[x 1,..., x n ]], where x 1,..., x n are independent variables. Let R be any ideal, p R a prime ideal and m = (x 1,..., x n ) the maximal ideal. First we prove that Lyubeznik numbers λ p,i (R/) are nothing but multiplicities of the characteristic cycle of the local cohomology modules Hm(H p n i (R)). Following the path given in the proof of G. Lyubeznik, we prove that the following multiplicities are also invariants of the ring R/: The multiplicities of the characteristic cycle of H n i (R). The multiplicities of the characteristic cycle of Hp p (H n i (R)). Among these multiplicities one may found: The Bass numbers µ p (p, H n i (R)). The Lyubeznik numbers λ p,i (R/) := µ p (m, H n i (R)). More precisely, in this chapter we prove the following results: Theorem Let A be a local ring which admits a surjective ring homomorphism π : R A, where R = k[[x 1,..., x n ]] is the formal power series ring. Set = ker π and let CC(H n i (R)) = m i,α T X α X, be the characteristic cycle of the local cohomology modules H n i (R). Then, the multiplicities m i,α depends only on A, i and α but neither on R nor on π. Collecting these multiplicities by the dimension of the corresponding irreducible varieties we obtain some other invariants. Even though these invariants are coarser, in some situations they will be better suited in order to make a precise description of the support of the local cohomology modules Definition Let R be an ideal. f CC(H n i (R)) = m i,α TX α X is the characteristic cycle of the local cohomology modules H n i (R) then we define: γ p,i (R/) := { m i,α dimx α = p}.

27 ntroduction xix These invariants have the same properties as Lyubeznik numbers. Namely: Proposition following properties: i) γ p,i (R/) = 0 if i > d. ii) γ p,i (R/) = 0 if p > i. iii) γ d,d (R/) 0. Let d = dimr/. The invariants γ p,i (R/) have the Then, we can collect these invariants in a triangular matrix that will be denoted by Γ(R/). On the other side, for the multiplicities of the characteristic cycle of the local cohomology modules Hp p (H n i (R)) we obtain the following result: Theorem Let A be a local ring which admits a surjective ring homomorphism π : R A, where R = k[[x 1,..., x n ]] is the formal power series ring. Set = ker π, let p A be a prime ideal and let CC(H p p (H n i (R))) = λ p,p,i,α T X α X, be the characteristic cycle of the local cohomology modules Hp p (H n i (R)). Then, the multiplicities λ p,p,i,α depends only on A, p, p, i and α but neither on R nor on π. As a consequence of this result, we prove that the Bass numbers, in particular the Lyubeznik numbers, are invariants of R/ since they are multiplicities of the characteristic cycle. More precisely: Proposition Let R be an ideal, p R a prime ideal and CC(H p p (H n i (R))) = λ p,p,i,α T X α X be the characteristic cycle of the local cohomology module Hp p (H n i (R)). Then, the Bass numbers with respect to p of H n i (R) are µ p (p, H n i (R)) = λ p,p,i,α p, where X α p is the subvariety of X = Spec (R) defined by p.

28 xx ntroduction Let k be a field of characteristic zero. n Chapter 3 we consider any of the following rings: R = k[[x 1,..., x n ]] the formal power series ring. R = k{x 1,..., x n } the convergent power series ring. R = k[x 1,..., x n ] the polynomial ring. The main result of this chapter is the computation of the characteristic cycle of the local cohomology modules H r (R) with support a monomial ideal R. First, we have to find among the tools introduced in Chapter 1, which one will be better suited for our purposes. A first approach is to use the Čech complex. Let = (xα 1,..., x α s ) be a squarefree monomial ideal. Consider the Čech complex Č : 0 R d 0 1 i 1 r 1 d R[ x α ] 1 d s 1 1 i R[ 1 x α 1 x α s ] 0 The characteristic cycle of the localizations R r := 1 R[ x α i 1 x α ir ] have been computed in Chapter 1. Then, since the cohomology of the Čech complex gives the local cohomology modules, i.e. H r(r) = Hr (Č ), we get CC(Hr (R)) = CC(Ker d r ) CC(m d r 1 ) by using the additivity of the characteristic cycle with respect to exact sequences. When the quotient ring R/ is Cohen-Macaulay, due to the fact that the cohomological dimension equals the height of the ideal, there is only a non vanishing local cohomology module (Proposition 3.1.1). Then, its characteristic cycle is easy to compute. Proposition Let R be an ideal of height h generated by squarefree monomials. f R/ is Cohen-Macaulay then: CC(H h (R)) = CC(R h) CC(R h+1 ) + + ( 1) s h CC(R s ) CC(R h 1 ) + + ( 1) h CC(R 0 ). Whether the quotient ring R/ is not Cohen-Macaulay, the computation of the characteristic cycles CC(Ker d r ) and CC(m d r ) is more involved due to the fact that the characteristic variety of the localizations R r := 1 R[ ] x α i 1 x α i r are not irreducible.

29 ntroduction xxi n the general case we will use the Mayer-Vietoris exact sequence. Basically, the process we will use is the following: Consider a presentation = U V of the ideal as the intersection of two simple ideals. Then, if we split the sequence H r U+V (R) H r U(R) H r V (R) H r (R) H r+1 U+V (R), into short exact sequences of kernels and cokernels 0 B r H r U(R) H r V (R) C r 0 0 C r H r (R) A r A r+1 H r+1 U+V (R) B r+1 0, we get that the characteristic cycle of the local cohomology module H r (R) is CC(H r (R)) = CC(C r ) + CC(A r+1 ) = = (CC(H r U(R) H r V (R)) CC(B r )) + (CC(H r+1 U+V (R)) CC(B r+1)). Note that we have reduced our problem to the computation of the characteristic cycles CC(HU r (R) Hr V (R)), CC(Hr+1 U+V (R)) and CC(B r) r. n order to compute the characteristic cycle of the local cohomology modules HU r (R), Hr V (R)) and Hr+1 U+V (R), we have to consider a decomposition of the ideals U, V and U + V. Then, we split the corresponding Mayer-Vietoris sequences in the same way we did before. Although we are working with some different sequences at the same time, the ideals U, V and U +V we are considering are simpler than ideal. We will repeat this process until we get a situation where one may compute the characteristic cycles CC(H r U (R) Hr V (R)), CC(H r+1 U+V (R)) and CC(B r) r that appear in all the Mayer-Vietoris sequences involved. n this chapter we give an inductive process that allows to choose the ideals U and V in a systematical way, i.e. independently of the ideal s complexity, by taking advantage of the good properties that satisfy the minimal primary decomposition = α1 αm of a squarefree monomial ideal. By means of this process, we reach a situation where the local cohomology modules H r U (R),

30 xxii ntroduction HV r (R)) and Hr+1 U+V (R) that appear in the different Mayer-Vietoris sequences are exactly one of the following 2 m 1 modules: H r αi1 + + αij (R), 1 i 1 < < i j m, j = 1,..., m. n order to organize this information, we introduce the partially ordered set (poset) = { 1, 2,..., m }, formed by all the sums of face ideals that appear in the minimal primary decomposition of, i.e. j := { αi1 + + αij 1 i 1 < < i j m}. Roughly speaking, we have broken the local cohomology module H r (R) into smaller pieces, the modules H r αi1 + + αij (R), that are labeled by the poset. These modules will be denoted initial pieces. The characteristic cycle of these pieces may be computed since R/ αi1 + + αij is Cohen-Macaulay. Then, it only remains to compute the characteristic cycle of the modules B r that appear in the different Mayer-Vietoris sequences. From the fact that the characteristic varieties of the initial pieces are irreducible, we prove that CC(B r ) is nothing but the sum of the characteristic cycles of local cohomology modules supported on sums of face ideals in the minimal primary decomposition of satisfying the following property: αij+1 Definition We say that αi1 + + αij j and αi1 + + αij + j+1 are paired if αi1 + + αij = αi1 + + αij + αij+1 Finally, once we control all the pieces of the Mayer-Vietoris sequences as well the kernels and cokernels, we use the additivity of the characteristic cycle with respect to exact sequences to compute CC(H r (R)). More precisely, CC(H r (R)) is the sum of the characteristic cycle of the initial pieces labeled by a subposet P, that it is calculated by means of an algorithm and is formed by the sums of face ideals αi1 + + αij such that are not paired and whose height is r + (j 1). Namely, if we consider the sets P j,r := { αi1 + + αij P j ht ( αi1 + + αij ) = r + (j 1)} then we have:

31 ntroduction xxiii Theorem Let R be an ideal generated by squarefree monomials and let = α1 αm be its minimal primary decomposition. Then : CC(H r (R)) = αi P 1,r CC(H r αi (R)) + + αi1 + αi2 P 2,r CC(H r+1 αi1 + αi2 (R)) + + CC(H r+(m 1) α1 + + αm (R)). α1 + + αm P m,r The information we obtain from the characteristic cycle of the local cohomology modules will be treated from different points of view. First, the varieties that appear in the characteristic cycle describe the support of the module H r (R). On the other side, the multiplicities allow us to describe some properties of the quotient ring R/ since they are invariants. Annhilation and support of local cohomology modules: A first reading of the formula for the characteristic cycle of the modules H r (R) allows us to obtain the following results: Annihilation of local cohomology modules. Proposition Cohomological dimension. Corollary Support of local cohomology modules. Proposition Krull dimension of local cohomology modules. Corollary Artinianity of local cohomology modules. Corollary We have to point out that these results are expressed in terms of the ideals in the minimal primary decomposition of the ideal. Arithmetical properties of the ring R/: The multiplicities of the characteristic cycle of local cohomology modules allow us to give some criteria to study the following properties: Cohen-Macaulay property. Propositions and Buchsbaum property. Propositions and Gorenstein property. Propositions and The type of Cohen-Macaulay rings. Proposition

32 xxiv ntroduction n the literature one may find some criteria to determine these properties, that use the topological properties of the Stanley-Reisner simplicial complex associated to the ring R/, see [86] for more details. Our criteria are based on the annihilation of certain multiplicities in the Cohen-Macaulay and Buchsbaum case. n the Gorenstein case we ask whether certain multiplicities are exactly 1. Anyway, they are expressed in terms of the ideals in the minimal primary decomposition of the ideal. Once we have computed the multiplicities of the characteristic cycle of the local cohomology modules we have seen that they are useful to describe properties of the modules H r (R) and the ring R/ as well. Now, we want to give an interpretation of these invariants and compare them with other known invariants. Combinatorics of the Stanley-Reisner rings and multiplicities: From the Stanley-Reisner correspondence, one may associate a simplicial complex to any squarefree monomial ideal R. n this section we give a description of the topological invariants described by the f-vector and the h-vector of in terms of the invariants d j B j := ( 1) i γ j,j+i (R/), i=0 i.e. the alternating sum of invariants γ p,i (R/) in a row of the matrix Γ(R/). Proposition Let R be a squarefree monomial ideal. The f-vector and the h-vector of the corresponding simplicial complex are described as follows: i) f k = d j=k+1 ( ii) h k = ( 1) k j k + 1 d k ( j=0 ) B j. d j k ) B j. By using these descriptions we also determine some invariants as the Euler characteristic of or the Hilbert series of R/ (Corollary ). n general, the invariants γ p,i (R/) are finer than the f-vector and the h-vector. They are equivalent whether R/ is Cohen-Macaulay (Corollary ).

33 ntroduction xxv A different interpretation of the multiplicities may be given by means of Alexander duality. Betti numbers and multiplicities: Let R be the Alexander dual ideal of a squarefree monomial ideal R. The Taylor complex T ( ) is a cellular free resolution with support on a simplicial complex whose vertices are labeled by the poset that we used in order to label the initial pieces of the local cohomology modules H r(r). Recall that the characteristic cycle of these modules is described by means of the poset P that we compute by using an algorithm. From the correspondence given by Alexander duality, this algorithm may be interpreted as an algorithm that allows to minimize the free resolution of given by the Taylor complex. Then, we can describe the multiplicities of the characteristic cycle CC(H r (R)) in terms of the Betti numbers of. Proposition Let R be Alexander dual ideal of a squarefree monomial ideal R. Then we have: β j,α ( ) = m n α +j, α (R/). n particular, the multiplicities m i,α of a fixed module H n i (R) describe the Betti numbers of the (n i)-lineal strand of. Whether R/ is Cohen- Macaulay there is only a non vanishing local cohomology, and we may recover the following result of J. A. Eagon and V. Reiner [20]. Corollary Let R be the Alexander dual ideal of a squarefree monomial ideal R. Then, R/ is Cohen-Macaulay if and only if has a linear free resolution. A generalization of that result expressed in terms of the projective dimension of R/ and the Castelnuovo-Mumford regularity of is given by N. Terai in [90]. We can give a different approach by using the previous results. Corollary Let R be the Alexander dual ideal of a squarefree monomial ideal R. Then we have: pd(r/) = reg ( ).

34 xxvi ntroduction Let k be a field of characteristic zero. n Chapter 4 we consider any of the following rings: R = k[[x 1,..., x n ]] the formal power series ring. R = k{x 1,..., x n } the convergent power series ring. R = k[x 1,..., x n ] the polynomial ring. The main result of this chapter is the computation of the characteristic cycle of the local cohomology modules Hp p γ (H r (R)) where R is a monomial ideal and p γ R is a face ideal. The techniques we will use throughout this chapter are a natural continuation of those used in the previous chapter. Namely, consider the short exact sequence 0 C r H r(r) A r+1 0, obtained by splitting the Mayer-Vietoris sequence H r U+V (R) H r U(R) H r V (R) H r (R) H r+1 U+V (R). Then, if we split the long exact sequence of local cohomology H p p γ (C r ) H p p γ (H r (R)) H p p γ (A r+1 ) H p+1 p γ (C r ), into short exact sequences of kernels and cokernels 0 Z p 1 H p p γ (C r ) X p 0 0 X p H p p γ (H r (R)) Y p 0 0 Y p H p p γ (A r+1 ) Z p 0 we get that the characteristic cycle of the local cohomology module H p p γ (H r (R)) is CC(H p p γ (H r (R))) = CC(X p ) + CC(Y p ) = = (CC(H p p γ (C r )) CC(Z p 1 )) + (CC(H p p γ (A r+1 )) CC(Z p )). Applying the long exact sequence of local cohomology to the sequences we have obtained in the process described in the previous chapter, we divide the local cohomology module Hp p γ (H r (R)) into smaller pieces, the modules Hp p γ (H r αi1 + + αij (R)) that are labeled by the poset P due to Theorem

35 ntroduction xxvii The characteristic cycle of these pieces may be computed by means of the Grothendieck s spectral sequence since R/ αi1 + + αij is Cohen-Macaulay (Proposition 4.1.1). More precisely, CC(H p p γ (H r αi1 + + αij (R)) = CC(H p+r p γ+ αi1 + + αij (R)). n order to organize the information given by these pieces we define the sets P γ,j,α := { αi1 + + αij P j p γ + αi1 + + αij = α }. Notice that it only remains to compute the characteristic cycle of all the modules Z p that appear in the different long exact sequences of local cohomology. The characteristic variety of these initial pieces is also irreducible, so we prove now that the characteristiccycle CC(Z p ) is nothing but the sum of characteristic cycles of the initial pieces whose corresponding sums of face ideals that appear in the minimal primary decomposition of satisfy the following property: For any face ideal α R we define the subsets P γ,j,α := { αi1 + + αij P j p γ + αi1 + + αij = α }. Definition We say that αi1 + + αij P γ,j,α and αi1 + + αij + αij+1 P γ,j+1,α are almost paired if ht ( αi1 + + αij ) + 1 = ht αi1 + + αij + αij+1 Finally, once we control all the pieces in the long exact sequences of local cohomology as well the kernels and cokernels, we use the additivity of the characteristic cycle with respect to exact sequences in order to compute CC(Hp p γ (H r(r))). More precisely, CC(Hp p γ (H r (R))) is the sum of the characteristic cycle of the pieces labeled by a poset Q P, that it is calculated by means of an algorithm and is formed by the sums of face ideals αi1 + + αij that are not almost paired and whose height is r + (j 1). Namely, if we consider the sets Q γ,j,r,α := { αi1 + + αij Q γ,j,α ht ( αi1 + + αij ) = r + (j 1)}, then we have:

36 xxviii ntroduction Theorem Let = α1 αm be the minimal primary decomposition of a squarefree monomial ideal R and p γ R a face ideal. Then: CC(H p p γ (H r (R))) = α {0,1} n λ γ,p,n r,α CC(H α α (R)), where λ γ,p,n r,α = Q γ,j,r,α such that α = p + (r + (j 1)). Whether p γ = m is the homogeneous maximal ideal we have the equality of sets P j,r = P α m,j,r,αm j, r. Applying the algorithm of cancellation of almost pairs and ordering by the number of summands and the height, we obtain the sets Q j,r = Q α m,j,r,αm. Then we get the following result: Corollary Let = α1 αm be the minimal primary decomposition of a squarefree monomial ideal R and m R be the homogeneous maximal ideal. Then: CC(H p m(h r (R))) = λ p,n r T X αm X, where λ p,n r = Q j,r such that n = p + (r + (j 1)). Among the information given by the characteristic cycle of these local cohomology modules we will focus on the multiplicities. More precisely, we will turn our attention to the invariants of the ring R/ described by the Bass numbers of the modules H r (R). Bass numbers and local cohomology modules: Let R be an ideal generated by squarefree monomials, p γ R a face ideal and m R the homogeneous maximal ideal. Recall that in Chapter 1 we have seen that the Bass numbers may be computed in the following way: Proposition Let R be an ideal generated by squarefree monomials and p γ R be a face ideal. f CC(H p p γ (H n i (R))) = λ γ,p,i,α T X α X, is the characteristic cycle of the local cohomology module Hp p γ (H n i (R)) then µ p (p γ, H n i (R)) = λ γ,p,i,γ. n particular, the Lyubeznik numbers are:

37 ntroduction xxix Corollary Let R be an ideal generated by squarefree monomials and m R be the homogeneous maximal ideal. f CC(H p m(h n i (R))) = λ p,i T X αm X, is the characteristic cycle of the local cohomology module Hm(H p n i (R)) then µ p (m, H n i (R)) = λ p,i. Once they are computed, we use the Bass numbers in order to describe more accurately the prime ideals that appear in the support of the modules H r (R) studied in the previous chapter. n particular we obtain the following results: Annihilation of Bass numbers. Proposition njective dimension of local cohomology modules. Corollary Associated primes of local cohomology modules. Proposition Small suport of local cohomology modules. Proposition Once again, we point out that the results are expressed in terms of the face ideals in the minimal primary decomposition of the ideal. To illustrate these computations we give some examples of modules H r (R) satisfying: id R H r (R) = dim RH r (R). H r (R) has no embedded primes. Min R (H r (R)) = Ass R(H r (R)). supp R (H r (R)) = Supp R(H r (R)). We also give examples of modules H r (R) satisfying: id R H r (R) < dim RH r (R). H r (R) has embedded primes. Min R (H r (R)) Ass R(H r (R)). supp R (H r (R)) Supp R(H r (R)).

38 xxx ntroduction Let A n k denote the affine space of dimension n over a field k, let X An k be an arrangement of linear subvarieties. Set R = k[x 1,..., x n ] and let R denote an ideal which defines X. n Chapter 5, we study the local cohomology modules H i (R) with special regard of the case where the ideal is generated by monomials. However, in this chapter we will pay more attention to the structure of these modules instead of their numerical invariants. Even though the tools we will use are independent of the characteristic of the field k, whether char(k) = 0 we will keep in mind the structure as D- module of the modules H r (R). On the other side, if char(k) > 0 we will use the notion of F -module introduced by G. Lyubeznik in [56, Definition 1.1]. We have to point out that an arrangement of linear varieties X determine a poset P (X) formed by the intersections of the irreducible components of X and the order given by the inclusion. For example, if X is defined by a squarefree monomial ideal R, P (X) is nothing but the poset defined in Chapter 3 identifying the sums of face ideals in the minimal primary decomposition of whether they describe the same ideal. First of all, in an analogous way to the construction of the Mayer-Vietoris spectral sequences for singular cohomology and l-adic cohomology introduced by A. Björner and T. Ekedahl [11], we prove the existence of a Mayer Vietoris spectral sequence for local cohomology: E i,j 2 = lim P (X) Hj p (R) H j i (R) (i) where p is an element of the poset P (X), p is the defining (radical) ideal of (i) the irreducible variety corresponding to p, and lim is the i-th left derived P (X) functor of the direct limit functor in the category of direct systems indexed by the poset P (X). Studying in detail this spectral sequence, we observe that the E 2 -term is defined by the reduced homology of the simplicial complex associated to the poset P (X) which has as vertices the elements of P (X) and where a set of vertices p 0,..., p r determines a r-dimensional simplex if p 0 < < p r. Proposition Let X A n k be an arrangement of linear varieties. Let K(> p) be the simplicial complex attached to the subposet {q P (X) q > p} of P (X). Then, there are R-module isomorphisms lim (i) P (X) Hj p (R) h(p)=j [ Hj p (R) k Hi 1 (K(> p); k) ],